I've commuted for years on assorted fixed gears. I never minded the steep angles, but I'm used to the toe-overlap that comes with most any track bike under a 56cm. You'll be on your own for hauling your kit and/or water, no eyelets. Fenders will be impossible to fit, unless you lace up a set of 650b wheels and then potentially be limited to ~35mm tires by the frame.
You can commute on anything, but this sounds like the kind of thing that's best saved for special road rides or sell it to someone who likes to go in circles.